Some Mathematical Studies on the 3D Incompressible MHD System and the Boltzmann Equation


Student thesis: Doctoral Thesis

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  • Xuanji JIA

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Awarding Institution
Award date7 Sep 2016


The thesis is concerned with the mathematical theories of two basic physical models from fluid mechanics and gas dynamics: the incompressible Magnetohydrodynamics (MHD) system and the Boltzmann equation. It is divided into two parts which are independent of each other.
In Part I, the Cauchy problem for the three dimensional incompressible MHD system is investigated. Some new regularity criteria for the global weak solutions are established in terms of the 3 × 3 mixture matrices and the 3D mixture vector involving only one velocity component. These results extend and improve the previous works in various ways and give some new understandings on the blowup mechanism of the MHD flow.
Part II is devoted to the asymptotic stability of non-equilibrium steady solutions to the exterior problem for the Boltzmann equation. For this problem, Ukai and Asano have gave the first rigorous but technical proof in [32], under the assumption that the temperature associating with the far-field Maxwellian and the one preserved by the kinetic boundary condition are the same. In [31], we generalize Ukai and Asano’s result in the sense that the two temperatures mentioned above are allowed to be different. The proof of the main theorem is based on the ideas developed in [44], [32] and [45] as well as some new observations.